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PHYSICAL REVIEW E 92, 050901(R) (2015)

Deterministic coherence resonance in coupled chaotic oscillators with frequency mismatch

A. N. Pisarchik1,2,* and R. Jaimes-Reategui2,3

1Centro de Investigaciones en Optica, Loma del Bosque 115, 37150 Leon, Guanajuato, Mexico2Computational Systems Biology, Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo,

28223 Pozuelo de Alarcon, Madrid, Spain3Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Dıaz de Leon, Paseos de la Montana, Lagos de Moreno,

Jalisco 47460, Mexico(Received 3 March 2015; revised manuscript received 3 July 2015; published 2 November 2015)

A small mismatch between natural frequencies of unidirectionally coupled chaotic oscillators can inducecoherence resonance in the slave oscillator for a certain coupling strength. This surprising phenomenon resembles“stabilization of chaos by chaos,” i.e., the chaotic driving applied to the chaotic system makes its dynamics moreregular when the natural frequency of the slave oscillator is a little different than the natural frequency of themaster oscillator. The coherence is characterized with the dominant component in the power spectrum of the slaveoscillator, normalized standard deviations of both the peak amplitude and the interpeak interval, and Lyapunovexponents. The enhanced coherence is associated with increasing negative both the third and the fourth Lyapunovexponents, while the first and second exponents are always positive and zero, respectively.

DOI: 10.1103/PhysRevE.92.050901 PACS number(s): 05.45.Xt, 05.45.Pq

It is common knowledge that a nonlinear system in thepresence of noise can exhibit resonance phenomena such asstochastic [1,2] or coherence [3] resonance. While the formeris seen as an optimal response to external periodic modulationwith respect to the noise intensity, the latter manifests itselfas increasing regularity in one of the system internal timescales without additional modulation. Coherence resonancewas detected first in excitable dynamical systems [3–7] andthen in bistable systems [8–10]. Examples are typically foundin biology in the form of neuron spiking dynamics [11].

Analogous phenomena were observed in deterministicchaotic systems without noise, since an external chaoticforcing acts in a similar way as noise. Deterministic stochasticresonance was found in bistable chaotic systems [12–14]and deterministic coherence resonance in chaotic systemswith delayed feedback [15–17] including diffusively coupledRossler oscillators [18]. In laser applications, an increase inthe injection diode current leads to an optimal regularity ofchaotic laser diode power dropouts [15,16].

It is not surprising that synchronization can affect chaoticdynamics [19]. Bragard et al. [20] observed chaos suppressionin chaotic oscillators with bidirectional asymmetric coupling.They found that adequate asymmetry and coupling betweentwo identical chaotic oscillators may force their dynamicstowards regular periodic oscillations. Since this phenomenonoccurs for the value of the coupling strength well below thevalue for complete synchronization, it was interpreted as ageneralized synchronization state.

In this Rapid Communication, we report on a significantlydifferent case of deterministic coherence resonance. Weconsider two unidirectionally coupled nonidentical chaoticoscillators, master x1 = F(x1,ω1) and slave x2 = F(x2,ω2) +σ (x1 − x2), where x1,2 are state variables of the master andslave systems, F is a vector function, and σ is a couplingstrength. The oscillators are only distinguished by their natural

*apisarch@cio.mx

frequencies ω1 and ω2. Due to nonlinearity, the dominantfrequency ω0 in the chaotic power spectrum of the masteroscillator usually does not coincide with its natural frequency.Since the master oscillator acts as a driving force for theslave oscillator, the dominant frequency of the slave oscillatoris entrained by the master oscillator when the coupling issufficiently strong, thus resulting in phase synchronization[21]. Recently, Pyragiene and Pyragas [22] showed that inthe phase synchronization state, the average phase difference〈φ2 − φ1〉 is negative if the frequency mismatch � = ω2 −ω1 < 0 and positive if � > 0. In the former case, the averageoscillators’ phases are locked with lag and in the latter casewith anticipation.

Here, we will show that a small frequency mismatch notonly leads to phase synchronization, but can also improvethe performance of chaotic oscillations inducing deterministiccoherence resonance in the slave system. This surprisingphenomenon resembles “stabilization of chaos by chaos,” i.e.,a chaotic system under a chaotic drive behaves more regular,almost periodic. The coherence is maximized with respect toboth the frequency mismatch � and the coupling strength σ .

Let us consider two unidirectionally coupled Rossleroscillators:

x1 = −ω1y1 − z1, x2 = −ω2y2 − z2,

y1 = ω1x1 + ay1, y2 = ω2x2 + ay2 + σ (y1 − y2), (1)

z1 = b + z1(x1 − c), z2 = b + z2(x2 − c).

The master oscillator is chaotic for a = 0.16, b = 0.1, c = 8.5,and ω1 = 1. The natural frequency of the slave oscillator ω2

and the coupling strength σ are used as control parameters. InFig. 1 we illustrate how a small mismatch � = 0.11 betweenthe natural frequencies of the master and slave oscillatorsenhances the coherence of the slave dynamics for the couplingstrength σ = 0.2. One can see that the dynamics of the slavesystem is more regular than that of the master oscillator.We should note that when the natural frequencies coincide(� = 0), the chaotic trajectories of the two oscillators areidentical.

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FIG. 1. (Color online) Coherence enhancement in coupledchaotic Rossler oscillators Eq. (1) with frequency mismatch � =0.11. Time series of (a) y1 and (b) y2 and phase portraits of (c) masterand (d) slave oscillators for coupling strength σ = 0.2.

Figure 2 shows the power spectra of the master and slaveoscillators. In the phase synchronization state, both oscillatorshave the same dominant frequency because the dominantfrequency of the slave oscillator is entrained by the masteroscillator. The coherence enhancement is characterized bystrong suppression of the broad chaotic spectrum of the slaveoscillator with simultaneous amplification of the periodiccomponent S0 at the dominant frequency ω0 = 1.05 lockedby the master oscillator.

Power spectrum and time series analyses provide significantinformation on the coherence (or regularity) of oscillations thatcan be quantitatively characterized by the following measures:(1) dominant spectral component S0, (2) normalized standarddeviation (NSD) of the peak amplitude (amplitude coherence),(3) NSD of the interpeak interval (IPI) (time coherence), and(4) Lyapunov exponents.

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FIG. 2. (Color online) Power spectra of master (red, light) andslave (blue, dark) oscillators for � = 0.11 and σ = 0.2.

FIG. 3. Resonance character of the maximum spectral componentS0 vs (a) natural frequency of the slave oscillator at σ = 0.2 and (b)coupling strength at ω2 = 1.11.

Although the difference between the dominant spectralcomponents of the master and slave oscillators is insignificant,the coherence resonance of the slave oscillator reveals itselfas an amplification of its dominant spectral component S0

as the control parameters (frequency mismatch and couplingstrength) change; the higher S0, the better is the regularity.This means that out of the resonance S0 is lower than at theresonance, while the powers of other spectral components arehigher. Figure 3(a) shows how S0 depends on ω2 while ω1 = 1.One can see a strong resonant amplification of S0 at ω0 = 1.05when ω2 approaches 1.11. Another (smaller) local maximumis observed at ω2 = 0.95. In fact, there are two coherenceresonances, however, the coherence for the lower resonancefrequency is worse than for ω2 = 1.11, but better than forω2 = 1 where the minimum in the dependence S0(ω2) occurs.The dependence of S0 on σ for fixed mismatch � = 0.11 alsodisplays a nonmonotonic character exhibiting a maximum atσ ≈ 0.2, as seen in Fig. 3(b). When the natural frequenciesof the master and slave oscillators coincide, they completelysynchronize and their power spectra are identical [red (light)line in Fig. 2].

In Fig. 4 we plot the maximum amplitude S0 of the slavepower spectrum as a function of the two control parameters,ω2 and σ . Phase synchronization is observed within the Arnoldtongue in the vicinity of ω2 = 1, where the dominant frequency

FIG. 4. (Color online) Dominant spectral component S0 at ω0 inthe (ω2,σ )-parameter space for coupled chaotic oscillators Eq. (1).

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FIG. 5. Frequency dependences of (a) y2 peak amplitude, (b) IPIof y2, (c) NSD of y2 peak amplitude, and (d) NSD of IPI for chaoticallydriven chaotic oscillator Eq. (1). σ = 0.2.

ω0 = 1.05 of the slave oscillator is entrained by the masteroscillator. The red (dark) spot inside this tongue means highercoherence.

Other characteristics of the coherence obtained from thetime series of the slave oscillator are shown in Fig. 5.The NSD of both the peak amplitude of y2 and the IPIminimize when ω2 = 1.11 for σ = 0.2. This is the signature ofcoherence resonance.

The stability of the system Eq. (1) is analyzed with theLyapunov exponent spectrum. All six Lyapunov exponentsλ1−6 in the (ω2,σ )-parameter space are plotted in Fig. 6. Thefirst Lyapunov exponent λ1 is always positive, which is theindicator of chaotic motion [Fig. 6(a)]. Depending on the con-trol parameters, the second exponent λ2 is either positive orzero [Fig. 6(b)], while the third and fourth exponents, λ3 andλ4, are either negative or zero. The latter exponents are ofspecial interest because they become negative in a certainrange of the control parameters, which can give importantinformation about stability of the slave system. In particular, asseen from Figs. 6(c) and 6(d), λ3 and λ4, being close to zero in awide range of control parameters, significantly decrease whenthe control parameters approach the region of ω2 ≈ 1.1 and0.14 < σ < 0.22 [central blue (dark) spots]. The decrease inthe Lyapunov exponents means an improvement in the stabilityof the chaotic attractor, resulting in a coherence enhancement.The fifth and sixth Lyapunov exponents, λ5 and λ6, do not giveus additional information because they are always negative[Figs. 6(e) and 6(f)].

The origin of the deterministic coherence resonance lies inthe nonlinear interaction of the chaotic forcing with systemnonlinearity. Due to nonlinearity, the dominant frequency ω0

differs from the natural frequency ω1; for the system Eq. (1)the frequency shift is δ = ω0 − ω1 = 0.05. Since the masteroscillator acts as a driving force for the slave oscillator, thedominant frequency of the slave oscillator is entrained by themaster oscillator when σ is sufficiently strong. This affects

FIG. 6. (Color online) Lyapunov exponents (a) λ1, (b) λ2, (c) λ3,(d) λ4, (e) λ5, and (f) λ6 of coupled system Eq. (1) in (ω2,σ )-parameterspace. Deterministic coherence resonance is associated with negativeλ3 and λ4 in the region of the central blue (dark) spots in (c) and (d).

the system stability that depends on both σ and �, as seenfrom Fig. 6. This dependence has a nonmonotonous charactershowing the highest stability for � ≈ 0.11 associated withminima in the negative Lyapunov exponents λ3 and λ4

[Figs. 6(c) and 6(d)]. The optimal σ corresponding to theminimal λ3 and λ4 grows up approximately linearly as ω2

increases. A similar resonance behavior is observed for othersystem parameters [23]. In the majority of cases, there is onlyone coherence resonance that occurs for |�| > |δ| (sgn � =sgn δ). However, for certain parameters another resonanceappears [23].

Interestingly, a periodic drive does not induce coherenceresonance in this system. Although the periodic force is known[24] to enhance the coherence of a chaotic system, it is not ableto induce coherence resonance. Next, we will demonstratethe effect of a periodic drive in the same chaotic system andcompare the results with the chaotic drive. Let us considerthe Rossler oscillator subject to harmonic force m sin(ωmt)applied to variable y:

x = −ωy − z,

y = ωx + ay + m sin(ωmt), (2)

z = b + z(x − c).

We take the modulation frequency ωm = 1.05 to be equal tothe dominant frequency ω0 of the master oscillator and themodulation amplitude m to be close to the amplitude of thechaotic drive σy1. The other parameters are kept the same asfor Eq. (1). Figure 7 shows the spectral component at the

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FIG. 7. (Color online) Dominant spectral component in the(ω,m)-parameter space for a chaotic oscillator with periodic driveEq. (2). ωm = 1.05.

dominant frequency as a function of both ω and m. Similar toFig. 4, the diagram in Fig. 7 exhibits a triangle-shaped Arnoldtongue of the frequency-locking range in the vicinity of ω = 1.However, inside this tongue we do not observe any coherenceresonance.

In Fig. 8 we plot the same characteristics as in Fig. 5, but forthe periodic drive with m = 2. One can see that both the NSDof peak y [Fig. 8(c)] and NSD of IPI [Fig. 8(d)] decrease almostmonotonically as ω increases, meaning monotonic coherenceenhancement, with a little exception for the periodic windowat ωm ≈ 0.81 [Figs. 8(a) and 8(b)].

However, this periodic regime differs from the coherenceresonance phenomenon observed in the coupled chaoticoscillators, where the coherence enhancement results from thesqueezing of the bifurcation diagrams (decreasing amplitudevariation) [Figs. 5(a) and 5(b)], but does not arise in a saddle-node bifurcation where the periodic window appears. Similarbehavior is observed for other m. These results indicate thatperiodic modulation is not able to induce coherence resonancein the chaotic system.

In conclusion, we have demonstrated the existence ofdeterministic coherence resonance in oscillations of a chaoticsystem unidirectionally coupled with another, almost identical,chaotic system in the presence of a small mismatch betweentheir natural frequencies. The improved coherence resembles“stabilization of chaos by chaos.” As counterintuitive as itmay seem, the two subsystems oscillate at the same dominantfrequency but follow a different dynamics; while the master os-cillator is chaotic, the slave oscillator is almost periodic. Using

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a paradigmatic example of Rossler oscillators, we have foundconditions and parameters where this surprising phenomenonoccurs. The coherence resonance has been identified withtime series and power spectra and quantitatively characterizedby the dominant spectral component, the minima of thenormalized standard deviation of both the peak amplitude andthe interpeak interval, as well as by the local minima in thethird and fourth Lyapunov exponents. Distinctly to the chaotic,a periodic drive of the same amplitude is not able to inducecoherence resonance. Therefore, the coherence enhancementoccurs due to an interaction of the chaotic systems, when theirphases synchronize and the dominant frequency of the slaveoscillator is entrained by the master oscillator.

The resonance behavior has been found for differentparameters of the Rossler attractors, as well as in a ringof three coupled oscillators with a frequency mismatch (theresults will be published somewhere). Therefore, we believethat the discovered phenomenon is general for a class ofunidirectionally coupled chaotic systems with pronounceddominant frequencies. Although a periodic drive can beof interest for some applications, for example, as a heartpacemaker, such systems are artificial, while coupled chaoticsystems are inherent to nature, and perhaps a similar effectmay occur in social, climate, or brain dynamics.

This work was supported by the BBVA Foundation(Spain) within the Isaac-Peral program of Chairs. R.J. ac-knowledges support from CONACYT (Mexico) (Project No.234594).

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